I can certainly understand your dissatisfaction with medieval depictions of heaven. However, your description of fun theory reminds me of the Garden of Eden. i.e. in Genesis 1-2, God basically says:
“I’ve created the two of you, perfectly suited for one another physically and emotionally, although the differences will be a world to explore in itself. You’re immortal and I’ve placed you in a beautiful garden, but now I’m going to tell you to go out and be fruitful and multiply and fill the earth and subdue it and have dominion over all living things; meaning build, create, procreate, invent, explore, and enjoy what I’ve created, which by the way is really really big and awesome. I’ll always be here beside you, and you’ll learn to live in perfect communion with me, for I have made you in my own image to love the process of creation as I do. But if you ever decide that you don’t want that, and that you want to go it alone, rejecting my presence and very existence, then there’s this fruit you can take and eat. But don’t do it, because if you do, you will surely die.”
It seems that the point of disagreement is that your utopia doesn’t have an apple. The basic argument of theodicy is that Eden with the apple is better than Eden sans apple. To the extent that free will is good, a utopia must have an escape option.
Or, to put it another way, obedience to the good is a virtue. Obedience to the good without the physical possibility evil is a farce.
It’s easy to look around and say, “How could a good God create THIS.” But the real question is, “How could a good God create a world in which there is a non-zero probability of THIS.”
Generic,
The y appears on both sides of the equation, so these are differential equations. To avoid confusion, re-write as:
(1) (d/dt) F(t) = A*F(t) (2) (d/dt) F(t) = e^F(t)
Now plug e^At into (1) and -ln(C-t) into (2), and verify that they satisfy the condition.